Finite Group Databasebeta

← #36381322

Subgroups — #36381322

Order 22599.

560 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31⋊H(5,𝔽3)C311yesmaximal, characteristic, complemented
2#21419405C311yesmaximal, characteristic, complemented
3#21419405C312yesmaximal, complemented
4#21419397C312yesmaximal, complemented
5#21419395C312yesmaximal, complemented
6C3×#14578895C311yesmaximal, characteristic, complemented
7C3×#14578893C311yesmaximal, characteristic, complemented
8(C3×C31⋊C3)⋊H(3,𝔽3)C312yesmaximal, complemented
9C31×#6911C311yesmaximal, characteristic, complemented
10C33×C31⋊C3C3211yescharacteristic, complemented
11C3×#11719689C3212yescomplemented
12C3×#11719689C3211yescharacteristic, complemented
13C3×C32⋊(C31⋊C3)—33no—
14C3×C32⋊(C31⋊C3)—36no—
15C3×C32⋊(C31⋊C3)C3212yes—
16C3×C32⋊(C31⋊C3)C3211yescharacteristic
17C3×C32⋊(C31⋊C3)C3212yescomplemented
18H(3,𝔽3)×C31⋊C3—36no—
19H(3,𝔽3)×C31⋊C3—33no—
20#14578899—33no—
21#14578899—36no—
22#14578895—36no—
23#14578895—33no—
24#14578893—36no—
25#14578893—33no—
26C3×3−1+2×C31C3211yescharacteristic, complemented
27C3×H(3,𝔽3)×C31C3211yescharacteristic
28C3×H(3,𝔽3)×C31C3212yescomplemented
29C31×#732—33no—
30C33×C31—31no—
31C33×C31—32no—
32C33×C31C3311yescharacteristic
33C32×C31⋊C3—32no—
34C32×C31⋊C3—39no—
35C32×C31⋊C3—31no—
36C32×C31⋊C3—31no—
37C32×C31⋊C3—31no—
38C32×C31⋊C3—32no—
39C32×C31⋊C3—93no—
40C32×C31⋊C3H(3,𝔽3)16yescomplemented
413−1+2×C31—31no—
423−1+2×C31—32no—
43H(3,𝔽3)×C31—32no—
44H(3,𝔽3)×C31—32no—
45H(3,𝔽3)×C31—32no—
46H(3,𝔽3)×C31—32no—
47H(3,𝔽3)×C31H(3,𝔽3)13yescomplemented
48C3×C9×C31—31no—
49C32⋊(C31⋊C3)—318no—
50C32⋊(C31⋊C3)—32no—
51C32⋊(C31⋊C3)—32no—
52C32⋊(C31⋊C3)—32no—
53C32⋊(C31⋊C3)—32no—
54C32⋊(C31⋊C3)—32no—
55C32⋊(C31⋊C3)—32no—
56C32⋊(C31⋊C3)—32no—
57C32⋊(C31⋊C3)—918no—
58C32⋊(C31⋊C3)H(3,𝔽3)13yescomplemented
59#11719689—32no—
60#11719689—32no—
61#11719689—32no—
62#11719689—31no—
63#11719689—32no—
64C3×C31⋊C9—32no—
65C3×C31⋊C9—31no—
66UT(4,𝔽3)—311nomaximal, Sylow
67C32×C31—33no—
68C32×C31—91no—
69C32×C31—96no—
70C32×C31—92no—
71C32×C31—271no—
72C32×C31C3×H(3,𝔽3)11yescharacteristic
73C32×C31C3×H(3,𝔽3)12yes—
74C32×C31C3×H(3,𝔽3)11yescharacteristic
75C3×C31⋊C3—36no—
76C3×C31⋊C3—36no—
77C3×C31⋊C3—33no—
78C3×C31⋊C3—94no—
79C3×C31⋊C3—91no—
80C3×C31⋊C3—93no—
81C3×C31⋊C3—93no—
82C3×C31⋊C3—96no—
83C3×C31⋊C3—96no—
84C3×C31⋊C3—92no—
85C3×C31⋊C3—271no—
86C3×C31⋊C3—279no—
87C9×C31—91no—
88C31⋊C9—92no—
89C31⋊C9—91no—
90H(5,𝔽3)—311no—
91#6911—311no—
92#6911—312no—
93#6911C31⋊C311yescharacteristic, complemented
94#6909—312no—
95#6909—312no—
96C3×AΣL(1,𝔽27)—311no—
97C3×AΣL(1,𝔽27)—311no—
98C32⋊H(3,𝔽3)—312no—
99C3×C31—32no—
100C3×C31—31no—
101C3×C31—31no—
102C3×C31—93no—
103C3×C31—272no—
104C3×C31C32⋊H(3,𝔽3)11yescharacteristic
105C31⋊C3—93no—
106C31⋊C3—96no—
107C31⋊C3—93no—
108C31⋊C3—272no—
109C34—311no—
110C3×3−1+2—312no—
111C3×3−1+2—311no—
112C3×3−1+2C3×C31⋊C311yescharacteristic, complemented
113C3×H(3,𝔽3)—311no—
114C3×H(3,𝔽3)—312no—
115C3×H(3,𝔽3)—312no—
116C3×H(3,𝔽3)—936no—
117C3×H(3,𝔽3)—936no—
118C3×H(3,𝔽3)—933no—
119C3×H(3,𝔽3)—933no—
120C3×H(3,𝔽3)C3×C31⋊C311yescharacteristic
121C3×H(3,𝔽3)C3×C31⋊C312yescomplemented
122#732—33no—
123#732—936no—
124#732—933no—
125AΣL(1,𝔽27)—936no—
126AΣL(1,𝔽27)—933no—
127AΣL(1,𝔽27)—936no—
128AΣL(1,𝔽27)—933no—
129C31UT(4,𝔽3)11yescharacteristic, complemented, Sylow
130C33—31no—
131C33—32no—
132C33—2793no—
133C33—316no—
134C33—939no—
135C33—931no—
136C33—932no—
137C33—931no—
138C33—932no—
139C33—931no—
140C33C32×C31⋊C311yescharacteristic
1413−1+2—31no—
1423−1+2—32no—
1433−1+2—931no—
1443−1+2—932no—
1453−1+2—932no—
1463−1+2—932no—
1473−1+2—932no—
148H(3,𝔽3)—32no—
149H(3,𝔽3)—32no—
150H(3,𝔽3)—32no—
151H(3,𝔽3)—32no—
152H(3,𝔽3)—27918no—
153H(3,𝔽3)—313no—
154H(3,𝔽3)—9318no—
155H(3,𝔽3)—932no—
156H(3,𝔽3)—932no—
157H(3,𝔽3)—932no—
158H(3,𝔽3)—932no—
159H(3,𝔽3)—932no—
160H(3,𝔽3)—932no—
161H(3,𝔽3)—932no—
162H(3,𝔽3)C32⋊(C31⋊C3)13yescomplemented
163C3×C9—31no—
164C3×C9—932no—
165C3×C9—931no—
166C32—33no—
167C32—91no—
168C32—96no—
169C32—92no—
170C32—2796no—
171C32—2793no—
172C32—2793no—
173C32—2794no—
174C32—2796no—
175C32—2791no—
176C32—2792no—
177C32—271no—
178C32—8371no—
179C32—8379no—
180C32—936no—
181C32—933no—
182C32—936no—
183C32H(3,𝔽3)×C31⋊C311yescharacteristic
184C32H(3,𝔽3)×C31⋊C312yes—
185C32C3×C32⋊(C31⋊C3)11yescharacteristic
186C9—91no—
187C9—2792no—
188C9—2791no—
189C3—32no—
190C3—31no—
191C3—31no—
192C3—93no—
193C3—2793no—
194C3—2796no—
195C3—2793no—
196C3—272no—
197C3—8372no—
198C3(C3×C31⋊C3)⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC3(C3×C31⋊C3)⋊H(3,𝔽3)
commutator, cosocleC33×C31C33
FrattiniC33C32×C31⋊C3
FittingC31×#6911C3
socleC3×C31C32⋊H(3,𝔽3)
hypercenter#6911C31⋊C3
hypocenterC31UT(4,𝔽3)
LCSG ▷ C33×C31 ▷ C3×C31 ▷ C31
UCSC1 ◁ C3 ◁ C33 ◁ #6911

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C31⋊H(5,𝔽3)C3semidirect
2. #21419405C3semidirect
3. #21419405C3semidirect
4. #21419397C3semidirect
5. #21419395C3semidirect
6. C3×#14578895C3semidirect
7. C3×#14578893C3semidirect
8. (C3×C31⋊C3)⋊H(3,𝔽3)C3semidirect
9. C31×#6911C3semidirect
10. C33×C31⋊C3C32semidirect
11. C3×#11719689C32semidirect
12. C3×#11719689C32semidirect
17. C3×C32⋊(C31⋊C3)C32semidirect
26. C3×3−1+2×C31C32semidirect
28. C3×H(3,𝔽3)×C31C32semidirect
40. C32×C31⋊C3H(3,𝔽3)semidirect
47. H(3,𝔽3)×C31H(3,𝔽3)semidirect
58. C32⋊(C31⋊C3)H(3,𝔽3)semidirect
93. #6911C31⋊C3semidirect
112. C3×3−1+2C3×C31⋊C3semidirect
121. C3×H(3,𝔽3)C3×C31⋊C3semidirect
129. C31UT(4,𝔽3)semidirect
162. H(3,𝔽3)C32⋊(C31⋊C3)semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C31⋊H(5,𝔽3)3
#214194053
#214193973
#214193953
C3×#145788953
C3×#145788933
(C3×C31⋊C3)⋊H(3,𝔽3)3
C31×#69113
UT(4,𝔽3)31